Algebraic Topology Code

So far I have only implemented cell complexes and in particular simpectial complexes as representations of topology. It is my intention to relate this to other representations of topology such as (co)homotopy groups and homology.

Cell Complexes

There are many ways to approach cell complexes. They originally arose from topology but they can also be used in a purely combinatorial context or an algebraic context.

We can also look at them as a greneralisation of the structure of a graph.

Graph:

Simpectial Complexes

In a graph an edge has arrows to two vertices (the source and target). Simpectial complexes generalise this to allow any number of vertices.

A 'simplex' is a set of vertices such that every subset is included in the simplex. For example a tetrahedron contains all its faces and lines.

A 'simplicial complex' is a set of these simplexes which may have vertices in common.

Because each simplex contains its subsets this is a combinatorial structure, when used in this way the structure is known as an 'abstract simplicial complex' optionally we can also associate a point in a space with each vertex, in this case the structure is known as an 'geometric simplicial complex' and it has geometric/topological/homotopy properties.

Homotopy

Homotopy Groups

The fundamental group of a pointed space (mostly represented here by simpectial complexes) is isomorphic to the fundamental group of the space (Seitert-Van Kampen theorem).

We look at loops of maps from a point to the given space.

Each loop, in the space, from the point back to itself is a generator for the group. These loops can then be composed.

The homotopy group is is finitely presented by generators and relations. This representation of a group is not, in general, algorithmically computable into other representations of a group.

The simplest homotopy group is the fundamental group. It determines when two paths, starting and ending the point, can be continuously deformed into each other. It records information about the shape in terms of holes in the topological space.

nth homotopy group. Instead of mapping a circle onto our topological space we map a sphere or higher order sphere onto the space.

To generate the fundamental group from the simplicial complex see the page here.

Homology

When we looked at the delta complex we got a chain of 'face maps' between each dimension and the next lower one.

Above is my attemp to draw, with each vertex appearing once, this makes the diagram very cluttered and hard to understand. Its easier to draw if nodes 4,5 and 6 appear twice as below:

Note: edges 4->5, 5->6 and 6->4 connect two faces but they wind in the same direction.

Where Are You

This is one of a series of pages describing code that I have written for FriCAS. Some of this code has already been included with the FriCAS library and so can be used as soon as FriCAS is installed (as described here). However the code I have put on Github may be more upto date.

The code I have written for algebraic topology is not incuded with FriCAS.

Theory

This page is about code I have implemented for algebraic topology structures. For a more theoretical explanation see the page here.